The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals, apparently inconsistent with the standard spin-density-wave scenario, has triggered proposals on the breakdown of the Kondo effect at the critical point. Here, we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo-Heisenberg model with volume-dependent Kondo coupling-this model displays both Kondo volume collapse and Kondo-breakdown transitions, as well as Lifshitz transitions associated with a change in the Fermi-surface topology. Within a large- N treatment, we find that the Lifshitz transition tends to merge with the Kondo volume collapse and hence becomes first order, whereas the Kondo breakdown transition remains of second order except for very soft lattices. Interesting physics emerges at the zero-temperature endpoint of the Kondo volume collapse: For electrons in two space dimensions, this endpoint is located at the Lifshitz line for a large range of parameters, thus two continuous quantum phase transitions coincide without fine tuning. We analyze the effective Landau theory for such a situation and discuss critical exponents. Finally, we relate our findings to current heavy-fermion experiments.